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Elliptic operators in subspaces and the eta invariant. (English) Zbl 1021.58017
Let $$A$$ be a self-adjoint elliptic operator on a closed manifold $$M$$ of dimension $$m$$. One defines $$\eta(s,A):=\sum_\nu sign(\lambda_\nu)|\lambda_\nu|^{-s}$$ as a measure of the spectral asymmetry of $$A$$. This has an analytic continuation to $$C$$; $$s=0$$ is a regular value and one sets $$\eta(A)=\eta(0,A)/2$$. If $$A_t$$ is a smooth $$1$$ parameter family of partial differential operators of order $$r$$ and $$r+m$$ is odd, then the mod $$Z$$ reduction of $$\eta(A_t)$$ is independent of the parameter $$t$$.
The authors derive a formula for this fractional part of the eta invariant in topological terms using the index theorem for elliptic operators on subspaces. The authors also study the $$K$$-theory of $$M$$ with coefficients in $$Z_n$$.
The authors have used the results of this paper in subsequent work to construct some new second order even order operators in odd dimensions with non-trivial eta invariant.

MSC:
 58J28 Eta-invariants, Chern-Simons invariants 58J20 Index theory and related fixed-point theorems on manifolds 58J22 Exotic index theories on manifolds 19K56 Index theory
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